A Comprehensive Study of Cepheid Variables in the Andromeda Galaxy Period Distribution, Blending, and Distance Determination

Home , Cepheid variable, Type II Cepheid

A&A 473, 847–855 (2007) Astronomy DOI: 10.1051/0004-6361:20077960 & c ESO 2007 Astrophysics

A comprehensive study of Cepheid variables in the Andromeda galaxy Period distribution, blending, and distance determination

F. Vilardell1,C.Jordi1,3, and I. Ribas2,3

1 Departament d’Astronomia i Meteorologia, Universitat de Barcelona, c/ Martí i Franquès, 1, 08028 Barcelona, Spain e-mail: [francesc.vilardell;carme.jordi]@am.ub.es 2 Institut de Ciències de l’Espai-CSIC, Campus UAB, Facultat de Ciències, Torre C5-parell-2a, 08193 Bellaterra, Spain e-mail: [email protected] 3 Institut d’Estudis Espacials de Catalunya (IEEC), Edif. Nexus, c/ Gran Capità, 2-4, 08034 Barcelona, Spain Received 28 May 2007 / Accepted 18 July 2007

ABSTRACT

Extragalactic Cepheids are the basic rungs of the cosmic distance scale. They are excellent standard candles, although their luminosities and corresponding distance estimates can be affected by the particular properties of the host galaxy. Therefore, the accurate analysis of the Cepheid population in other galaxies, and notably in the Andromeda galaxy (M 31), is crucial to obtaining reliable distance determinations. We obtained accurate photometry (in B and V passbands) of 416 Cepheids in M 31 over a five year campaign within a survey aimed at the detection of eclipsing binaries. The resulting Cepheid sample is the most complete in M 31 and has almost the same period distribution as the David Dunlap Observatory sample in the Milky Way. The large number of epochs (∼250 per filter) has permitted the characterisation of the pulsation modes of 356 Cepheids, with 281 of them pulsating in the fundamental mode and 75 in the first overtone. The period-luminosity relationship of the fundamental mode Cepheids has been studied and a new approach has been used to estimate the effect of blending. We find that the blending contribution is as important as the metallicity correction when computing Cepheid distance determinations to M 31 (∼0.1 mag). Since large amplitude Cepheids are less affected by blending, we have used those with an amplitude AV > 0.8 mag to derive a distance to M 31 of (m − M)0 = 24.32 ± 0.12 mag. Key words. Stars: variables: Cepheids – stars: distances – galaxies: individual: M 31 – galaxies: distances and redshifts – methods: observational

1. Introduction (Udalski et al. 1999; Mochejska et al. 2001; Udalski et al. 2001; Pietrzynski´ et al. 2004; Macri et al. 2006), obtaining large sam- Cepheids are probably the most studied variable stars. Their ples of Cepheids with accurate photometry. The detailed study large amplitudes and intrinsic luminosities make them eas- of the observed Cepheids has emphasized the importance of an ily detectable in most photometric variability surveys. In ad- issue that was usually overlooked in most photometric stud- dition, their well-known period-luminosity (P-L) relationship ies: the effect of blending. It has been proposed (Mochejska (Leavitt & Pickering 1912) has made these variable stars one et al. 2000, hereafter MMSS00) that the magnitude of Cepheids of the main cornerstones in deriving extragalactic distances (see may be affected by the light of unresolved companion stars Freedman et al. 2001, for an historical review). The importance (i.e., blends). The effect of blending is somewhat different from of Cepheids for distance determination stands in contrast with crowding or confusion noise, since companion stars appear to be the relative lack of additional information on the specific char- in the same point-like source. Therefore, even when achieving acteristics of extragalactic Cepheids and the possible corrections a perfect point-spread function modeling, blending could still because of their particular properties (i.e., metallicity). A clear be present. The effect can be the same as in spectroscopic bi- example is the Andromeda galaxy (M 31), where the first iden- naries, where the individual components cannot usually be re- tification of Cepheids was already performed by Hubble (1929). solved from ground-based images. After the observations of Baade & Swope (1965, and references therein), few efforts have been dedicated to further analyze the When studying extragalactic Cepheids the spatial resolu- Cepheid population in M 31. tion decreases linearly with the distance of the host galaxy This trend has changed in recent years with the emer- and, as a consequence, the number of possible blends increases. gence of new observational capabilities. Several variability sur- Small blending values are expected for the Large Magellanic veys have started to study the stellar content in M 31 (Macri Cloud (LMC), where individual Cepheids can be resolved from 2004, and references therein) and other Local Group galaxies neighboring stars, even from ground-based observations. The situation changes in M 31 and M 33, where mean blending . . Based on observations made with the Isaac Newton Telescope op- values of 0 18 mag and 0 16 mag, respectively, have been ob- erated on the island of La Palma by the Isaac Newton Group in the tained (Mochejska et al. 2004, hereafter MMSS04). These re- Spanish Observatorio del Roque de los Muchachos of the Instituto de sults would imply, when extrapolated to more distant galaxies, Astrofísica de Canarias. a downward revision of the Hubble constant between 5% to

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20077960 848 F. Vilardell et al.: A comprehensive study of Cepheid variables in the Andromeda galaxy

10% (as explained by Gibson et al. 2000). By contrast, Bresolin 14 et al. (2005) found an upper limit on blending in NGC 300 (at 15 ∼2Mpc)of0.04 mag. In addition, Gibson et al. (2000) showed ﬀ that the systematic e ect of blending on farther galaxies (be- 16 tween 4 and 25 Mpc) is almost negligible. Therefore, results obtained so far seem to indicate that blending has an important 17 contribution when observing Local Group galaxies and dimin- ishes when observing distant galaxies. Gibson et al. (2000) ex- 18 plained such behavior as a consequence of the background levels

in M 31 and M 33 (with a large number of stars detected around V 19 Cepheid variables) not being representative of the more distant galaxies (where the background levels are the result of several 20 unresolved sources). Because of the importance of the subject, a comprehensive study of the effect of blending on Cepheid dis- 21 tance determinations is highly desirable. One of the most recent variability surveys in M 31 was car- 22 ried out by us and was centered on the North-Eastern quadrant 23 of the galaxy (Vilardell et al. 2006, hereafter Paper I). Our main goal was to detect eclipsing binaries, although a sample 24 of 416 Cepheids also resulted from the reduction and analysis -0.5 0 0.5 1 1.5 2 2.5 3 process. The large number of detected stars and the high quality B-V of the resulting light curves have allowed the current study of Fig. 1. Color–magnitude diagram for 37 241 objects with photometric Cepheid properties in M 31. errors below 0.1 mag in the reference catalog of Paper I (grey dots) and the 416 identified Cepheids (black circles). Objects above the dashed line are saturated. Error bars on the left indicate the mean error, in mag- 2. Photometric sample nitude and color, at different magnitude values. The photometric survey in Paper I was carried out with the 2.5 m Isaac Newton Telescope (INT) at La Palma (Spain) over shallow surveys while, on the other hand, the distribution of the the course of five observing runs (between 1999 and 2003). observations can make the identification of long period Cepheids An implementation of the difference image analysis technique difficult. (Wozniak 2000) was used in the reduction process to automati- To evaluate the observational biases, the 416 Cepheids cally detect 3964 variable stars with ∼250 observations per band from Paper I were compared with the 420 Cepheids in M 31 (B and V). The analysis of variance algorithm (Schwarzenberg- of the GCVS (Samus et al. 2004). Both samples have al- Czerny 1996) was used to identify periodic variable stars. The most the same number of stars, but the two period distribu- folded light curves were visually inspected, which led to the tions (Fig. 2) are found to be largely different, as demonstrated identification of 437 eclipsing binaries and 416 Cepheids. by the Kolmogorov-Smirnov tests, which provide values of The photometric measurements were transformed to the stan- around 10−13. The origin of the observed difference can be ex- dard system through the observation of several Landolt (1992) plained by two main factors. Firstly, the Paper I sample is known star fields, obtaining magnitude errors between 0.03 and to have a bias for the longest period Cepheids because of the 0.11 mag. Finally, the phase-weighted mean magnitudes (Saha observational window function (see Paper I for further infor- & Hoessel 1990) of the Cepheid sample were determined for mation). And secondly, the GCVS presents an important bias each passband (B and V). Figure 1 shows the color-magnitude at short periods because the GCVS is shallower than our survey diagram of the observed stars with good photometric precision. and faint Cepheids are missing. Furthermore, both Cepheid sam- Cepheid variables are highlighted. ples in M 31 can be compared with the MW sample of the David Dunlap Observatory1 (DDO, Fernie et al. 1995). As can be seen (Fig. 2), the period distributions of Paper I and the DDO sample 3. Period distribution are very similar (with a Kolmogorov-Smirnovtest value of 0.49). Only a slight difference is observed at long periods, possibly be- The period distribution of Cepheids in Local Group galaxies has cause of the observational bias in the Paper I sample. Therefore, been a major source of debate. Evolution models predict a dis- as expected, equivalent period distributions are obtained for sim- placement on the peak of the period distribution as a function ilar galaxies, including the secondary peak at P > 10 d in metal of the metallicity of the host galaxy (Becker et al. 1977). It has rich galaxies. also been observed that the Milky Way (MW) period distribution The conspicuous similarity between the period distributions displays a dip at around 10 days, while such a feature is missing of MW and M 31 becomes even more striking when com- in more metal-poor galaxies such as the LMC. Several theories pared with a galaxy of different metallicity (Fig. 2). More than have been put forward to explain the bimodal distribution, but a 1300 Cepheids have been observed in the LMC as part of the satisfying scenario is still lacking (see Antonello et al. 2002, for OGLE II survey2 (Udalski et al. 1999). The resulting period dis- an extended discussion). tribution reveals a single peak, which, as expected, is shifted to- Since the MW and M 31 have similar morphological type wards shorter periods with respect to those of the distributions and chemical composition, the observed period distributions are also expected to be similar. The results obtained so far have 1 Data obtained from: http://www.astro.utoronto.ca/DDO/ prevented a direct comparison because of observational biases, research/cepheids/cepheids.html which are often difficult to evaluate. On the one hand, faint 2 Data obtained from: ftp://sirius.astrouw.edu.pl/ogle/ Cepheids (with usually short periods) can be missing in some ogle2/var_stars/lmc/cep/catalog/ F. Vilardell et al.: A comprehensive study of Cepheid variables in the Andromeda galaxy 849

0.5 0.1 Paper I sample GCVS sample 0.08 0.4

V 0.3

0.06 21 R

0.04 0.2

Fraction of Cepheids 0.1 0.02 0.0 0 0.5 0.1 Paper I sample DDO sample 0.4 0.08 OGLE LMC sample

B 0.3 21

0.06 R 0.2 0.04 0.1 Fraction of Cepheids 0.02 0.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 log(P) -0.5 0 0.5 1 1.5 2 2.5 3 log(P) 6.0

Fig. 2. Top: normalized period distribution for the 416 detected 5.0 Cepheids in Paper I compared with the 420 Cepheids in M 31 of the 4.0

GCVS. Bottom: normalized period distribution for the 416 detected V 21

Cepheids in M 31 compared with the 509 Cepheids in the MW from ϕ 3.0 the DDO sample. The OGLE LMC Cepheid sample is also shown for 2.0 comparison. 1.0

0.0 of the MW and M 31. Therefore, the great similarity between the 6.0 DDO and Paper I samples indicates that both the observational 5.0 biases and period distributions are equivalent. 4.0 B 21

ϕ 3.0

4. Fourier decomposition 2.0 It is well known that Fourier decomposition of the Cepheid 1.0 light curves can provide valuable information on the real na- 0.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ture of these variable stars, especially on their pulsation modes log(P) (Beaulieu et al. 1995). The procedure involves fitting the coeffi- Fig. 3. Fourier coefficients in each passband (B and V) as a function cients (Ak and ϕk)ofaFourierseriesoftheform: of period. The pulsation mode of the 315 Cepheids is identified: black J circles – fundamental mode; grey circles – first overtone. m(t) = A0 + Ak cos (2πkΦ(t) + ϕk) (1) k=1 where m(t)andΦ(t) are the magnitude of the Cepheid and the phase at time t, respectively. Once the best coefficients have been obtained, the pulsation mode of the Cepheid variable can be de- χ2 = / ff the fit to correctly reproduce its shape, even when . . . is rel- rived from the amplitude ratio Rk1 Ak A1 and phase di erence ffi d o f ϕ = ϕ − kϕ . atively low. Problematic fits usually have coe cients with large k1 k 1 error bars. Therefore, coefficient errors were computed (accord- A prerequisite of the Fourier decomposition of the ing to Petersen 1986) and light curves with σ > 0.01 mag, 416 Cepheids identified in M 31 is the determination of the maxi- A0 σ > 0.1, or σϕ > 0.6rad(0.2π rad) were also rejected. mum order of the fit, J. In agreement with other studies in the lit- R21 21 erature (Antonello et al. 1990; Beaulieu et al. 1995; Zakrzewski The above criteria yielded a sample of 315 Cepheids with et al. 2000), we observed that an iterative solution provided sat- accurate Fourier fits in both passbands (B and V). The R21 isfactory results. Therefore, for each light curve, the fitting pro- and ϕ21 values in both passbands (Fig. 3) have been used = χ2 cedure started with J 1 and increased until the value of d.o.f. to classify 75 Cepheids pulsating in the first-overtone mode varied by less than one. Although the Fourier fit always provided (FO) and 240 pulsating in the fundamental mode (FM). Since values for all the light curves, several important aspects were FO Cepheids are not expected to have periods longer than considered before studying the results. Firstly, since measure- ∼7 days, the FM sample has been completed with all Cepheids ments with large photometric errors can provide unreliable fits, with log P > 0.9(8 days), regardless of the quality of their light curves with a mean photometric error larger than 0.1 mag Fourier fit. The final sample of 281 FM Cepheids includes three χ2 > were excluded. Secondly, all light curves with d.o.f. 7were type II Cepheid candidates, but they are easily identified as rejected to eliminate the fits that did not accurately match the a consequence of the detailed analysis of the P-L distribution observations. Finally, large gaps in a light curve can prevent (Sect. 5.1). 850 F. Vilardell et al.: A comprehensive study of Cepheid variables in the Andromeda galaxy

18 5.1. Absorption 19 The eﬀect of diﬀerential absorption can be partially corrected from the observed Cepheid color (B − V) through the color- 20 excess (E(B − V) ≡ (B − V) − (B − V)0, Feast 1999). We used

V 3 21 P-L relationships from Udalski et al. (1999) to estimate B0 and V0 values. The LMC distance modulus was assumed to be 22 (m − M)0 = 18.42 ± 0.06 mag and was computed from the weighted mean of all the LMC distance determinations with 23 eclipsing binaries (Fitzpatrick et al. 2003). The reason for us- 19 ing LMC relationships (instead of those for the MW), was mo- tivated by several recent results. First, Gieren et al. (2005) sug- 20 gested that the MW Cepheid distances, mostly obtained through the Baade-Wesselink method, could be aﬀected by a systematic 21

B bias when converting the observed radial velocities into pulsation velocities. Second, Macri et al. (2006) found that LMC P-L 22 slopes provide a better agreement with the P-L distribution of 23 Cepheids in NGC 4258 than MW relationships. Finally, the new parallax measurements of several MW Cepheids (Benedict et al. 24 0 0.5 1 1.5 2 2007; van Leeuwen et al. 2007) yield P-L slopes that seem to log(P) be in better agreement with the LMC than with previous MW relationships. Fig. 4. Observed magnitudes as a function of period for 356 Cepheids. After obtaining the color excess for each Cepheid, a total-to- Filled black circles: fundamental mode Cepheids with accurate Fourier R ≡ / − = . ± . coefficients. Empty black circles: fundamental mode Cepheids with- selective extinction ratio of V A(V) E(B V) 3 1 0 3 out accurate Fourier coefficients. Grey circles: first overtone mode (Fitzpatrick 1999) was used to compute the absorption and the pulsators. V0 magnitude for every Cepheid. The resulting P-L distribution can be compared with the LMC P-L relationship (Fig. 5), assuming a M 31 distance modulus of (m − M)0 = 24.44 ± 0.12 mag (Ribas et al. 2005). An offset between the V0 values and the LMC P-L relationship is clearly observed, but the effects of metallicity 5. Period-luminosity relationship and blending have still to be considered. Finally, as previously mentioned, three of the studied The P-L diagram for the 356 classified Cepheids (281 FM and Cepheids seem to be of type II. To further investigate these ob- 75 FO) reveals a large scatter in both passbands (Fig. 4), espe- jects they have been kept in the studied sample of FM Cepheids cially in B. The origin of the observed scatter can be understood (Sects. 5.3 and 6). if we consider that the field of view covers about 7 kpc at the distance of M 31 ((m − M)0 = 24.44 ± 0.12 mag, Ribas et al. 2005). The large covered area in M 31, the fact that Cepheids 5.2. Metallicity are located along the spiral arms, and the clumpy structure of As explained above, the zero point of the P-L relationship may the interstellar medium (clearly observed in the survey images), ff ff depend on metallicity. To account for this e ect, the metallicity indicates that di erential absorption in the disk is most likely of each Cepheid needs to be estimated, and we did so by con- responsible for an important fraction of the observed scatter. In sidering a galactocentric metallicity dependence. According to addition, the larger dispersion of the P-L distribution in B com- Zaritsky et al. (1994), the M 31 galactocentric metallicity depen- pared to V reinforces the interstellar absorption hypothesis. dence can be modeled by: Another possible effect on the observed scatter is metallicity. Several authors have suggested (see, i.e., Gieren et al. 2005) that 12 + log (O/H) = (9.03 ± 0.09) − (0.28 ± 0.10)(ρ/ρ0 − 0.4) the slope of the Cepheid P-L relationship is basically indepen- ρ ρ dent of metallicity. However, comparison of the tip of the red gi- where is the de-projected galactocentric radius and 0 is the ant branch and Cepheid distance determinations to several galax- isophotal radius (77.44 arcmin for M 31). Following the procedure in Baade & Arp (1964), the Cepheid galactocentric radius ies has shown that a slight dependence on metallicity is present ◦ (Sakai et al. 2004). Therefore, a metallicity dependence on the was obtained assuming a position angle of 38 , an inclination of 12◦.5 (Simien et al. 1978), and an M 31 center position of zero point of the Cepheid P-L relationship could exist. Since ◦ metallicity within M 31 decreases as a function of the galac- α = 00h42m44s.31 δ =+41 16 09. 4 (in J2000.0 coordinates, tocentric distance (Zaritsky et al. 1994), some of the observed Cotton et al. 1999). scatter could also be introduced by metallicity differences. The metallicity dependence of the Cepheid P-L relationship ff In addition, the Cepheid measured photometry can be af- and the galactocentric metallicity variation implies that a di er- fected by blends. The mean seeing of our images is around ent P-L relationship must be used for each individual Cepheid when computing the value V0 − MV . Alternatively, one can com- 1 arcsec. Therefore, each observed point-like source corresponds − to ∼4 pc at the distance of M 31. Since Cepheids are usually lo- pute V0 MV by using a universal P-L relationship and then applying a correction for each Cepheid a posteriori. According cated in young star clusters and associations, it is likely that the − observed magnitude of a Cepheid can be the combination of sev- to Sakai et al. (2004), the correction on the computed V0 MV eral unresolved sources. 3 Updated relationships were obtained from the OGLE II web site: Below we analyse the three aforementioned effects that are ftp://sirius.astrouw.edu.pl/ogle/ogle2/var_stars/lmc/ the probable sources of the scatter in the P-L diagram. cep/catalog/README.PL F. Vilardell et al.: A comprehensive study of Cepheid variables in the Andromeda galaxy 851

intrinsic amplitude. The observed amplitude and the blended 18 mean magnitude m can be expressed as: A = m − M (5) m + M 19 m = (6) 2 where m and M are the observed magnitudes at minimum and 20 maximum light of a Cepheid, respectively. In the same way, the intrinsic amplitude and the blending-free mean magnitude mi

0 can be expressed as: V 21 Ai = mi − Mi (7) mi + Mi mi = · (8) 22 2 From Eqs. (5)–(8) the difference on the mean magnitude of a given Cepheid as a consequence of blending can be expressed as: 23 A−Ai ∆=m−mi = (m − mi) − · (9) 2 24 0 0.5 1 1.5 2 Considering that (m − mi) can be defined as: log(P) f + f f Fig. 5. Absorption-corrected V magnitude as a function of period for m − mi = −2.5log c = −2.5log 1 + (10) 281 fundamental mode Cepheids. Filled circles: stars with accurate fc fc Fourier fit. Empty circles: stars without accurate Fourier fits. Black line: Udalski et al. (1999) P-L relationship, for a distance modulus to M 31 and isolating f / fc from (4), the difference on the mean magni- of 24.44 mag (Ribas et al. 2005). Grey lines: distance corrected P-L tude (9) can be expressed as: relationships for type II Cepheids as given by Alcock et al. (1998). 100.2A − 10−0.2A ∆=2.5log · (11) 100.2Ai − 10−0.2Ai value is δ(m − M)0/δ(O/H) = −0.25 ± 0.09 mag/dex when using the Udalski et al. (1999) P-L relationships. Therefore, we com- Therefore, the variation on the mean magnitude of a given puted metallicity corrections by comparing the assumed metal- Cepheid because of blending can be computed from the ampli- licity of each M 31 Cepheid with the metallicity of the LMC tude of the Cepheid. Although Ai and ∆ are unknown quantities, (12 + log (O/H) = 8.5 dex, Sakai et al. 2004). The resulting the above equation can be solved in combination with period- corrections for the studied sample lie between −0.15 mag and color and P-L relationships. The results shown below are based −0.05 mag. on the Udalski et al. (1999) relationships, but almost identical results were obtained with other LMC relationships (Sandage et al. 2004). 5.3. Blending The next step of the process involves recalling that mm, The best approach to study the effect of blending given our where m is the phase-weighted intensity-average mean magni- available data sets is by using the observed Cepheid amplitudes tude of the observed Cepheids. Therefore, defining the blending- (Antonello 2002). The intrinsic amplitude Ai of a Cepheid is free color excess as: given by the expression: i − i ≡ i − i − − E(B V ) (B V ) (B V)0 (12) f Ai = −2.5log c (2) the blended color excess can be expressed as: Fc i i E(B − V) = E(B − V ) +∆B − ∆V (13) where fc and Fc are the fluxes at minimum and maximum light, i i respectively. When blending is present, the observed amplitude where ∆B = B − B and ∆V = V − V . Analogously, the distance A can be expressed in the following form: modulus can be expressed as: fc + f (V − M ) = (m − M) +∆ −R (∆ − ∆ ) (14) A = −2.5log (3) 0 V 0 V V B V F + f c where: where f is the sum of fluxes of all blending sources. From these − = −R − − two equations, one can write: (V0 MV ) V V E(B V) MV (15) i i i ⎛ ⎞ (m − M)0 = V −RV E(B − V ) − MV . (16) . Ai f ⎜100 4 + ⎟ A = . ⎜ fc ⎟ · The combination of Eqs. (14) and (11) reveals that the value of 2 5log⎝ f ⎠ (4) 1 + (V − M ) is only a function of the amplitude of the Cepheids fc 0 V and the distance modulus to M 31. In addition, the fact that i Since A and f / fc are always positive quantities, the observed the color excess is obtained from the observed color of the amplitude of a blended Cepheid is always smaller than its Cepheids (which is affected by blending) introduces the color 852 F. Vilardell et al.: A comprehensive study of Cepheid variables in the Andromeda galaxy

3 2 2.5 1.8 2

1.6 B

A 1.5

1.4 1

0.5 1.2

B 0

A -2 -1 0 1 2 1 ∆ B 2 0.8

1.5 0.6 V

0.4 A 1

0.2 0.5

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 A 0 V -2 -1 0 1 2 ∆ V Fig. 6. Amplitude relationship for the 240 fundamental mode Cepheids. Fig. 7. Observed amplitude of the 240 fundamental mode Cepheids with accurate Fourier fits as a function of the computed variation on term (∆B − ∆V ) in Eq. (14). Therefore, and contrary to the in- tuitive interpretation, when reddening is computed from the ob- the mean magnitude due to blending. Lines of constant intrinsic am- − plitude from 0.5 to 2 or 2.5 mag in steps of 0.5 mag are also shown served color of Cepheids (B V) the blended distance modulus (dashed lines). can be either larger or smaller than the intrinsic distance modulus (m − M)0, depending on the values of (∆B − ∆V ). Considering that all Cepheids are roughly at the same dis- Given the large number of studied Cepheids, the obtained tance, (m − M)0 = 24.44 ± 0.12 mag (Ribas et al. 2005), and as- blending values can also be used to determine a mean blending Ai Ai value for Cepheids in M 31. The results are shown in Table 1 suming that V and B are linearly dependent (Fig. 6), Eq. (14) can be numerically solved. In fact, when the intrinsic amplitude where they are also compared with previous blending determi- of the Cepheid tends to zero, the observed amplitudes have to be nations in M 31 and M 33. The values in Paper I were obtained zero in all passbands. Therefore, both amplitudes were consid- from the third light contribution in eclipsing binary systems, and Ai = αAi the MMSS00 and MMSS04 values were obtained from compari- ered to be proportional (i.e., B V ). The assumption that Cepheid amplitudes, when observed at different passbands, are son of Hubble Space Telescope (HST) and ground based images. proportional can be considered a first-order approximation of the The blending factors (S ) presented in MMSS00 and MMSS04 Fourier coefficient interrelations (Ngeow et al. 2003). were transformed into variation on the mean magnitude values by considering that: Since the interrelations between B and V have not been accurately worked out, we computed the proportionality factor em- f pirically from the observed amplitude of Cepheids. The ampli- ∆=−2.5log 1 + ≡−2.5log(1 + S ) (17) f tude of the studied Cepheids can be reliably estimated from the c Fourier fits (Sect. 4). Therefore, the 240 FM Cepheids with accu- where fc is the flux of the Cepheid at mean magnitude. rate Fourier fits were used to compute the amplitude proportion- The observed difference with MMSS00 could be due to the ality factor (Fig. 6), obtaining a mean value (with 2.5σ clipping) assumed distance moduli or to the observing conditions. On Ai /Ai A /A = . ± . of B V B V 1 435 0 011. the one hand, the reported uncertainties on the LMC and M 31 The above distance modulus to M 31 and the amplitude pro- distance moduli could produce a variation of 0.12 mag in ∆B portionality factor, when applied to Eq. (14), yielded the intrin- and ∆V . On the other hand, blending depends on seeing condi- sic amplitudes of the Cepheids. Once the intrinsic amplitudes tions and background level. Large seeing images increase the are known, the difference on the mean magnitude can be com- blending and high signal-to-noise data enables the detection of puted (Fig. 7). The large resulting uncertainties in ∆B and ∆V faint stars, decreasing the computed background and increasing clearly indicate that not much can be said from the individual the blending contribution. Considering that similar results have blending of each Cepheid. The uncertainties were obtained from been obtained for the eclipsing binary sample (which comes a Monte Carlo run with 1000 realizations on the input parame- from the same observational data but from a completely different ters of each Cepheid. The positive ∆ values (implying negative procedure), the obtained differences with the values reported in blending) provide additional information on the associated un- MMSS00 could be the result of different observing conditions. certainties (probably of the order of 0.1–0.2 mag). In any case, The method used by MMSS00 can only provide lower limits to it is interesting to observe that moderate intrinsic amplitudes are blending values, since the HST point spread function (used as obtained for the entire sample, as it can be deduced from the a blending-free reference) could still hide unresolved compan- lines of constant intrinsic amplitude in Fig. 7 (at ∆=0, A = Ai). ions. This is especially the case for the B-band results, where the Furthermore, the Cepheid with ∆V > 1 mag is the only suspected available HST data was of low signal-to-noise, thus implying type II Cepheid with an accurate Fourier fit (see Fig. 5). The un- that only the most severe blends were detected. In addition, the realistic blending value provides an additional evidence in favor relatively small sample used by MMSS00 cannot be considered of the type II classification. representative of the Cepheid population in M 31. F. Vilardell et al.: A comprehensive study of Cepheid variables in the Andromeda galaxy 853

Table 1. Blending values in M 31 and M 33 obtained from several methods. Top: mean values and their errors. Bottom: median values.

Reference Sample Number S V S B∆V ∆B [mag] [mag] MMSS00 Cepheids in M 31 22 (10 in B)0.19 ± 0.03 0.06 ± 0.06 −0.18 ± 0.03 −0.05 ± 0.05 Paper I Eclipsing binaries 48 0.31 ± 0.07 0.30 ± 0.06 −0.23 ± 0.05 −0.25 ± 0.04 Present work FM Cepheids 240 0.31 ± 0.03 0.37 ± 0.04 −0.23 ± 0.02 −0.24 ± 0.03 Present work FM Cepheids with P > 12 days 37 0.14 ± 0.06 0.20 ± 0.10 −0.09 ± 0.05 −0.10 ± 0.07 Present work FM Cepheids with AV > 0.8mag 37 0.10 ± 0.05 0.11 ± 0.07 −0.06 ± 0.04 −0.05 ± 0.06

MMSS04 Cepheids in M 33 95 (57 in B)0.24 ± 0.03 0.29 ± 0.06 −0.20 ± 0.02 −0.23 ± 0.04 MMSS04 Cepheids in M 33 with P > 10 days 60 (39 in B)0.16 ± 0.04 0.20 ± 0.05 −0.14 ± 0.025 −0.17 ± 0.04 Reference Sample Number Median(S V )Median(S B)Median(∆V )Median(∆B) [mag] [mag] MMSS00 Cepheids 22 (10 in B)0.12 0.00 −0.12 0.00 Paper I Eclipsing binaries 48 0.09 0.16 −0.09 −0.16 Present work FM Cepheids 240 0.20 0.15 −0.19 −0.16 Present work FM Cepheids with P > 12 days 37 0.09 0.04 −0.09 −0.05 Present work FM Cepheids with AV > 0.8mag 37 0.09 0.04 −0.09 −0.05

MMSS04 Cepheids in M 33 95 (57 in B)0.13 0.15 −0.13 −0.15 MMSS04 Cepheids in M 33 with P > 10 days 60 (39 in B)0.07 0.10 −0.07 −0.10

It is interesting to note that the results of MMSS04, based on 6. Distance determination a relatively large sample of Cepheids in M 33, have mean values similar to ours (although slightly different median values; As previously mentioned, long period and large amplitude Table 1). Such good agreement should be taken with caution as Cepheids are less affected by blends. Therefore, when com- the host galaxies, the methods used, the period distributions of puting the mean distance modulus to M 31, a systematic trend the samples, and therefore the associated systematic errors are should be observed for increasing period and amplitude cuts different. Nevertheless, it is encouraging that estimates made us- (i.e., rejecting short period or low amplitude Cepheids). The V0 − ing different methods in two somewhat differing spiral galaxies MV values obtained after the metallicity correction (Sect. 5.2) yield similar results. were used to compute weighted mean distance determinations to M 31 (with 2.5σ clipping). The resulting values assume a − = . ± . To further analyze the effects of blending we have applied distance modulus to the LMC of (m M)0 18 42 0 06 several cuts to the FM sample. It has traditionally been ar- mag (Sect. 5.1). The amplitudes for the 41 FM Cepheids with- gued that the longer period (and brighter) Cepheids should be out accurate fits were computed from the observations and were less affected by blending (Macri et al. 2006). We computed the also included in order to obtain a better coverage in periods and mean blending for Cepheids with a period longer than 12 days amplitudes. and obtained, as expected, lower blending values (Table 1). Figure 8 shows that the distance modulus increases as the Furthermore, considering that blending decreases the Cepheid minimum period or minimum amplitude cuts increase. We ob- amplitude, larger amplitude Cepheids should also be less af- serve that the distance modulus value stabilizes for AV > 0.8 fected by blending and, in fact, the 37 Cepheids with AV > mag. The most likely cause for the observed tendency is that 0.8 mag do have lower blending values (Table 1). blending is lower than photometric errors (or even zero) for large amplitudes. The most suitable period cut is more difficult Finally, a blending-corrected color excess can be computed to compute, although an increasing tendency is also observed. from Eq. (13), obtaining a mean value of E(Bi − Vi) = 0.305 ± The observed behavior is explained if long period Cepheids are ff 0.011. The blended mean color excess is slightly lower (E(B − still a ected by large blends, thus still introducing a bias in the V) = 0.296 ± 0.012), indicating that blending sources are bluer derived distance modulus. Hence, an amplitude cut is the best than Cepheids. The difference is more significant when consid- choice to derive the distance to M 31 from the studied Cepheid ering only Cepheids with large blending values (∆ < −0.5mag sample. From this analysis (Fig. 8), the 66 Cepheids with am- V A > . and ∆B < −0.5 mag), obtaining a blending-corrected color ex- plitude V 0 8 mag seem to represent the best sample for cess of E(Bi − Vi) = 0.472 ± 0.028, whereas E(B − V) = distance determination. 0.315 ± 0.030. Therefore, considering the color-magnitude dia- The effect of removing small amplitude Cepheids is clearly gram (Fig. 1) and the distribution of Cepheids along the spiral visible in Fig. 9, where the P-L diagram for the largest amplitude arms, it is likely that early-type and young main sequence stars Cepheids is shown. Two stars with AV > 0.8 mag are placed far are responsible for the large measured blending. The obtained below the general P-L distribution (empty circles in Fig. 9). Both results are in good agreement with the results in MMSS04, who stars are on the type II Cepheids relationships, reinforcing the also found that M 33 blending sources were on average bluer hypothesis that these stars are, in fact, type II Cepheids and they than the Cepheids. To further check this scenario we used young have not been considered from now on. Finally, the obtained P-L MW open clusters to compute the light contribution that the slope, with a value of −2.83 ± 0.12 mag dex−1,isclosertothe main sequence stars would introduce on a Cepheid variable in Udalski et al. (1999) LMC value of −2.779 ± 0.031 mag dex−1 −1 the cluster. The resulting predictions on ∆V and ∆B − ∆V are in than to MW P-L slopes (e.g.: −3.087±0.085 mag dex , Sandage good agreement with the inferred blending values in M 31. et al. 2004), therefore favoring the adopted P-L relationships. 854 F. Vilardell et al.: A comprehensive study of Cepheid variables in the Andromeda galaxy

24.4

24.35 20 V -M 24.3 0 0 V V 21

24.25 22

24.2 23

24.15 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 A 24 log(P) V 0 0.5 1 1.5 2 log(P) Fig. 8. Weighted mean V0 − MV values for the 281 fundamental mode Cepheids in M 31 for different cuts. Error bars indicate the error on the Fig. 9. Absorption-corrected V magnitude as a function of period for mean. The dashed line represents the adopted distance determination 281 fundamental mode Cepheids. Empty grey circles: complete fun- A > of (m − M) = 24.32 ± 0.12 mag. Left: each distance determination damental mode sample. Filled black circles: Cepheids with V 0 . A > . includes all Cepheids with period longer than the specified value. Right: 0 8 mag. Empty black circles: two stars with V 0 8 mag excluded each distance determination includes all Cepheids with amplitude larger from the distance determination. Black line: Udalski et al. (1999) P-L than the specified value. relationship, for a distance modulus to M 31 of 24.44 (Ribas et al. 2005) and a mean metallicity correction of −0.1 mag (Sect. 5.2). Grey lines: P-L relationships for type II Cepheids as given by Alcock et al. (1998) at distance of M 31. From the considerations above, the M 31 distance modulus obtained from the studied sample of Cepheids is (m − M) 0 variables, resulting in 281 FM Cepheids. The posterior analysis (V − M )A> . = 24.32 ± 0.12 mag. This value is compat- 0 V 0 8 revealed that at least three of these stars are type II Cepheids. ible with most distance determinations found in the literature The analysis of the P-L relationship for the FM Cepheids re- and the assumed distance modulus in Sect. 5.3. Considering veals a large scatter, which is not explained solely through the that some large amplitude Cepheids may still be affected by effects of interstellar absorption and metallicity. Although addi- blends, the distance modulus we derive could have a slight neg- tional efforts are needed to reduce the obtained uncertainties, a ative bias. Note that our final value is 0.09 mag larger than new method to compute the effect of blending is presented. The the weighted mean distance modulus for the 281 FM Cepheids exact dependence of the amplitudes among different passbands, ((V − M ) = 24.23 ± 0.12 mag). Therefore, blending is clearly 0 V as well as more precise amplitude determinations, would greatly an important effect that has to be considered when obtaining ex- improve the results shown here. Even by considering the large tragalactic distance determinations. In fact, blending has nearly uncertainties, the large number of studied Cepheids provides an the same impact on the final distance determination for M 31 as accurate characterization of the mean blending values. We con- the metallicity correction. clude that the most likely cause of blending seems to be the light from unresolved stars belonging to the same stellar associations or clusters as Cepheid variables. 7. Summary The effect of blending has been shown to be larger than As a result of a variability survey in the North-Eastern quadrant 0.09 mag in the distance modulus to M 31, thus having an ef- of M 31 (Paper I), 416 Cepheids were detected and measured fect as important as the metallicity correction. Therefore, blend- with a large number of epochs (∼250) per filter (B and V). The ing should always be taken into account when obtaining extra- large number of detected Cepheids has allowed the direct study galactic distance determinations with Cepheids. It has also been of their period distribution. The results reveal that the period dis- shown that an amplitude cut is more adequate for obtaining un- ff tributions in M 31 and the MW are extremely similar and that biased distance determinations than a period cut, since the e ect the Cepheid sample obtained is almost as complete as the DDO of blending is better accounted for. Finally, the large amplitude MW sample. The only difference may be the lack of long period Cepheids have been used to obtain a distance modulus to M 31 − = . ± . Cepheids. of (m M)0 24 32 0 12 mag. The large number of epochs obtained in both filters per- Acknowledgements. The authors are very grateful to L. M. Macri for useful mits an accurate pulsation mode identification for 240 FM and comments during the preliminary stage of this work. Thanks are also due to 75 FO Cepheids. Although some FO Cepheids were previously the anonymous referee for providing valuable comments and information on ad- detected in M 31 (Fliri et al. 2006), our sample represents an ditional blending determinations. The David Dunlap Observatory and the OGLE teams are also acknowledged for making all their data publicly available in important increase in the number of detected FO pulsators and FTP form. This program was supported by the Spanish MCyT grant AyA2006- opens a new window to study the basic properties of these stars 15623-C02-01/02. F.V. acknowledges support from the Universitat de Barcelona in another Local Group galaxy. The sample of FM Cepheids was through a BRD fellowship. I.R. acknowledges support from the Spanish MEC completed with 41 long period (i.e., longer than 8 days) Cepheid through a Ramón y Cajal fellowship. F. Vilardell et al.: A comprehensive study of Cepheid variables in the Andromeda galaxy 855

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